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23Models - A collection of potentially useful models A...

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A collection of potentially useful models We’ve already seen two very common mixed models: for subsampling for designed experiments with multiple experimental units Here are three more general classes of models Random coefficient models, aka multi-level models Models for repeated experiments Models for repeated measures For complicated problems, may need to combine ideas c 2011 Dept. Statistics (Iowa State University) Stat 511 section 23 1 / 26 Random coefficient models A regression where all coefficients vary between groups Example: Strength of parachute lines. Measure strength of a parachute line at 6 positions Physical reasons to believe that strength varies linearly with position Model by y i = β 0 + β 1 X i + i , where X is the position, y is the strength, and i indexes the measurement What if have 6 lines, each with 6 observations? Measurements nested in line suggests: Y ij = β 0 + β 1 X ij + ν j + ij = ( β 0 + ν j ) + β 1 X ij + ij , where j indexes the line. Intercept varies between lines, but slope does not c 2011 Dept. Statistics (Iowa State University) Stat 511 section 23 2 / 26 1 2 3 4 5 6 900 1000 1100 1200 1300 Position Strength Parachute line strength c 2011 Dept. Statistics (Iowa State University) Stat 511 section 23 3 / 26 Random coefficient regression models allow slope to also vary Y ij = ( β 0 + α j 0 ) + ( β 1 + α j 1 ) X ij + ij u = [ α 10 , α 11 , α 20 , α 21 , . . . , α 60 , α 61 ] u = 1 1 0 0 . . . 0 1 2 0 0 . . . 0 1 3 0 0 . . . 0 1 4 0 0 . . . 0 1 5 0 0 . . . 0 1 6 0 0 . . . 0 0 0 1 1 0 . . . 0 . . . . . . . . . . . . . . . 0 0 0 1 6 36 × 12 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 23 4 / 26
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α j 0 α j 1 N 0 0 , G G = σ 2 0 σ 01 σ 01 σ 2 1 sometimes see model written as: Y ij = β 0 j + β 1 j X ij + ij , β j 0 β j 1 N β 0 β 1 , G R usually assumed σ 2 e I . Σ = ZGZ + R is quite complicated, can write out but not enlightening. Features: Var Y ij not constant, depends on X ij , even if R is σ 2 e I Cov Y ij , Y i j not constant, depends on X ij and X i j Cov Y ij , Y i j = 0 , since obs. on different lines assumed independent so Σ is block diagonal, with non-zero blocks for obs. on same line c 2011 Dept. Statistics (Iowa State University) Stat 511 section 23 5 / 26 Customary to include a parameter for the covariance between intercept and slope random effects. if omit, then model is not invariant to translation of X i.e., fixed effect part of the regression is the same model even if shift X , e.g. X 3 . random effects part is the same only if include the covariance some parameter values will change if X shifted, but structure stays the same. Can extend model in at least two ways: 1. More parameters in regression model e.g. quadratic polynomial: Y ij = β 0 j + β 1 j X ij + β 2 j X 2 ij + ij , Example: Allan Trapp’s MS. Longevity of stored seed, quadratic, 2833 seed lots of maize, each with 3 to 7 observations. 2. More levels of random effects 6 Measurements per line, 4 lines per parachute Measurements nested within Lines, Lines nested within Chutes c 2011 Dept. Statistics (Iowa State University) Stat 511 section 23 6 / 26 Repeated Experiments In some scientific fields, it is expected that you will repeat the entire experiment Plant Pathology, Agronomy. Often journals will not publish unless
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